Improved Radiation and Combustion Routines for a
Large Eddy Simulation Fire Model
KEVIN McGRATTAN, JASON FLOYD, GLENN FORNEY and HOWARD BAUM
Building and Fire Research Laboratory
National Institute of Standards and Technology (NIST)
Gaithersburg, Maryland, 20899, USA
SIMO HOSTIKKA
VTT Building and Transport
Espoo, Finland
ABSTRACT
Improvements have been made to the combustion and radiation routines of a large eddy
simulation fire model maintained by the National Institute of Standards and Technology.
The combustion is based on a single transport equation for the mixture fraction with state
relations that reflect the basic stoichiometry of the reaction. The radiation transport equation is solved using the Finite Volume Method, usually with the gray gas assumption for
large scale simulations for which soot is the dominant emitter and absorber. To make the
model work for practical fire protection engineering problems, some approximations were
made within the new algorithms. These approximations will be discussed and sample calculations presented.
KEY WORDS: finite volume method, large eddy simulation, mixture fraction model
INTRODUCTION
In cooperation with the fire protection engineering community, a large eddy simulation fire
model, Fire Dynamics Simulator (FDS), is being developed at NIST to study fire behavior
and to evaluate the performance of fire protection systems in buildings. Version 1 of FDS
was publicly released in February 2000 [1, 2]. To date, about half of the applications of
the model have been for design of smoke handling systems and sprinkler/detector activation
studies. The other half consist of residential and industrial fire reconstructions. Throughout
its early development, FDS had been aimed primarily at the first set of applications, but
following the initial release it became clear that some improvements to the fundamental
algorithms were needed to address the second set of applications. The two most obvious
needs were for better combustion and radiation models to handle large fires in relatively
small spaces, like scenarios involving flashover.
An improved version of FDS, called FDS 2, was released in the fall of 2001. The low Mach
number Navier-Stokes equations of FDS version 1 and their numerical solution based on
large eddy simulation generally remain the same in FDS 2. What is different are the comFIRE SAFETY SCIENCE--PROCEEDINGS OF THE SEVENTH INTERNATIONAL SYMPOSIUM, pp. 827-838
Copyright © International Association for Fire Safety Science
827
bustion and radiation routines. FDS 1 contains a relatively simple combustion model that
utilizes “thermal elements,” massless particles that are convected with the flow and release heat at a specified rate. While this model is easy to implement and relatively cheap
computationally, it lacks the necessary physics to accommodate underventilated fires. A
method that handles oxygen consumption more naturally includes an equation for a conserved scalar quantity, known as the mixture fraction, that tracks the fuel and product gases
through the entire combustion process. The model assumes that the reaction of fuel and
oxygen is infinitely fast, an appropriate assumption given the limited resolvable length and
time scales of most practical simulations.
Radiation transport in FDS version 1 was based on a simple algorithm whereby a prescribed fraction of the fire’s energy was distributed on surrounding walls according to a
point source approximation. The fire itself was idealized as a discrete set of Lagrangian
particles, referred to as “thermal elements,” that released convective and radiative energy
onto the numerical grid. This method had two major problems. The first was that only the
fire itself radiated; there was no wall to wall or gas to gas radiative heat transfer. Second,
the method became expensive when the fire began to occupy a large fraction of the space.
A better method for handling radiative heat transfer is to return to the fundamental radiation
transport equation for a non-scattering gray gas. The equation is solved using techniques
similar to those for convective transport in finite volume methods for fluid flow, thus the
name given to it is the Finite Volume Method (FVM).
The new combustion and radiation routines allow for calculations in which the fire itself
and the thermal insult to nearby objects can be studied in more detail than before when
the fire was merely a point source of heat and smoke. Studies have been performed to
examine in detail small scale experiments like the cone calorimeter [3], and fundamental
fire scenarios like pool fires [4, 5] and small compartment fires [6]. These calculations
are finely resolved, with grid cells ranging from a few millimeters to a few centimeters.
However, the majority of model users are still interested mainly in smoke and heat transport
in increasingly complex spaces. The challenge to the model developers is to serve both the
researchers and practitioners with a tool that contains the appropriate level of fire physics
for the problem at hand. This paper describes how the model was improved to better
describe the combustion and radiation phenomena, while at the same time maintain its
robust hydrodynamic transport routines to handle large scale smoke movement problems.
COMBUSTION
The simplest combustion model that includes the basic stoichiometry of the reaction assumes that the fuel, the oxygen, and the combustion products can be related to a single
conserved quantity called the mixture fraction. The obvious advantage of the mixture fraction approach is that all of the species transport equations are combined into one, reducing
the computational cost. The mixture fraction combustion model is based on the assumption
that large scale convective and radiative transport phenomena can be simulated directly, but
physical processes occurring at small length and time scales must be represented in an approximate manner. In short, the model adopted here is based on the assumption that the
combustion occurs much more rapidly than the resolvable convective and diffusive phe828
nomena. These same assumptions were made in FDS version 1, where Lagrangian particles known as “thermal elements” were used to represent small packets of unburned fuel.
The problem with this idea was that the elements were pre-programmed to release their
energy in a given amount of time; the time being derived from flame height correlations of
well-ventilated fires. The method broke down when the fire became under-ventilated, or
even if the fire was pushed up against a wall [4].
The mixture fraction, Z, is a conserved quantity representing the fraction of gas at a given
point that was originally fuel. The mass fractions of all of the major reactants and products
can be derived from the mixture fraction by means of “state relations,” empirical expressions arrived at by a combination of simplified analysis and measurement. Start with a
general reaction between a fuel and oxygen:
νO [O] + νF [F] → ∑ νP [P]
(1)
P
The numbers νO , νF and νP are the stoichiometric coefficients for the overall combustion
process that reacts fuel “F” with oxygen “O” to produce a number of products “P”. The stoichiometric equation (1) implies that the mass consumption rates for fuel, ṁ000
F , and oxidizer,
ṁ000
,
are
related
as
follows:
O
ṁ000
ṁ000
O
F
=
(2)
νF MF
νO MO
Under these assumptions, the mixture fraction Z is defined as:
sYF − YO −YO∞
νO MO
Z=
; s=
I
∞
νF MF
sYF +YO
(3)
By design, it varies from Z = 1 in a region containing only fuel to Z = 0 where the oxygen
mass fraction takes on its undepleted ambient value, YO∞ . Note that YFI is the fraction of fuel
in the fuel stream. The quantities MF , MO , νF and νO are the fuel and oxygen molecular
weights and stoichiometric coefficients, respectively. The mixture fraction Z satisfies the
conservation law
∂ρZ
+ ∇ · ρuZ = ∇ · ρD∇Z
(4)
∂t
where ρ is the gas density, D is the material diffusivity, and u is the flow velocity. Equation (4) is a linear combination of the fuel and oxygen mass conservation equations. The
assumption that the chemistry is “fast” means that the reaction that consumes fuel and oxygen occurs so rapidly that the fuel and oxygen cannot co-exist. The interface between fuel
and oxygen is the “flame sheet” defined by
Z(x,t) = Z f
;
Zf =
YO∞
sYFI +YO∞
(5)
Because the mixture fraction is a linear combination of fuel and oxygen, additional information is needed to extract the mass fractions of the major species from the mixture
fraction. This information comes in the form of “state relations.” Relations for the major
components of a simple one-step hydrocarbon reaction are given in Fig. 1.
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FIGURE 1: State relations for propane.
An expression for the local heat release rate can be derived from the conservation equation
for the mixture fraction and the state relation for oxygen. The starting point is Huggett’s
relationship for the heat release rate as a function of the oxygen consumption rate
q̇000 = ∆HO ṁ000
O
(6)
Here, ∆HO is the heat release rate per unit mass of oxygen consumed, an input parameter
to the model that is usually on the order of 13,000 kJ/kg [7]. Equation (6) is the basis for
oxygen-consumption calorimetry, and it is consistent with the assumption of infinite-rate
kinetics. The oxygen mass conservation equation
ρ
DYO
= ∇ · ρD∇YO + ṁ000
O
Dt
(7)
can be transformed into an expression for the local heat release rate using the conservation
equation for the mixture fraction (4) and the state relation for oxygen YO (Z).
dYO
dYO
d 2YO
000
−ṁO = ∇ · ρD
∇Z −
∇ · ρD∇Z = ρD
|∇Z|2
(8)
dZ
dZ
dZ 2
Neither of these expressions for the local oxygen consumption rate is particularly convenient to apply numerically because of the discontinuity of the derivative of YO (Z) at Z = Z f
for the ideal state relations. However, an expression for the oxygen consumption rate per
unit area of flame sheet can be derived from Eq. (8)
dYO ρD ∇Z · n
(9)
−ṁ00O =
dZ Z<Z f
In the numerical algorithm, the local heat release rate is computed by first locating the
flame sheet, then computing the local heat release rate per unit area, and finally distributing
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this energy to the grid cells cut by the flame sheet. In this way, the ideal, infinitessimally
thin flame sheet is smeared out over the width of a grid cell, consistent with all other gas
phase quantities.
RADIATION
Version 1 of FDS had a simple radiation transport algorithm that used randomly chosen
rays from energy-carrying Lagrangian particles to walls and other solid obstructions. This
method has two major problems. The first is that only the fire itself radiates; there is no
wall to wall or gas to gas radiative heat transfer. Second, the method becomes expensive
when the fire begins to occupy a large fraction of the space. A better method for handling
radiative heat transfer is to consider the Radiative Transport Equation (RTE) for a nonscattering gas
s · ∇Iλ (x, s) = κ(x, λ) [Ib (x) − I(x, s)]
(10)
where Iλ (x, s) is the radiation intensity at wavelength λ, Ib (x) is the source term given
by the Planck function, s is the unit normal direction vector and κ(x, λ) is the absorption
coefficient at a point x for wavelength λ.
For practical simulations, the spectral dependence cannot be resolved accurately. Instead,
the radiation spectrum can be divided into a relatively small number of bands, and a separate RTE derived for each one [3]. However, even with a small number of bands, the
solution of the RTEs is very time consuming. Fortunately, in most large scale fire scenarios
soot is the most important combustion product controlling thermal radiation from the fire
and hot smoke. Because the radiation energy is distributed over a wide range of wavelengths under these circumstances, it is convenient to assume that the gas behaves as a gray
medium. The spectral dependence is lumped into one absorption coefficient and the source
term is given by the blackbody radiation intensity
Ib (x) = σ T (x)4 /π
(11)
In optically thin flames, where the amount of soot is small compared to the amount of CO2
and water, the gray gas assumption may produce significant over-predictions of the emitted
radiation, in which case the multi-band radiation model is needed.
For the calculation of the gray (and if necessary band) mean absorption coefficients κ (κn ),
a narrow-band model, RadCal [8], is combined with FDS. At the beginning of a simulation,
the absorption coefficients are tabulated as a function of mixture fraction and temperature.
During the simulation the local absorption coefficient is found from a pre-computed table.
An important consideration in computing the entries in the table is the fact that the radiation
spectrum is dependent on a path length due to the widening and overlapping of the individual lines. Thus the “effective” absorption coefficient will be a function of the distance
over which the line-of-sight form of the RTE is integrated. In FDS, a fraction of the characteristic length of the computational domain is chosen as the path length used by RadCal
in computing effective gray gas absorption coefficients. A proper definition of path length
in the context of band mean absorption coefficients is still a subject of active research.
In calculations of limited spatial resolution, the source term, Ib , in the RTE requires special
831
treatment in the neighborhood of the flame sheet because the temperature is smeared out
over a grid cell and thus is considerably lower than one would expect in a diffusion flame
if the cell is relatively large. Moreover, the soot volume fraction within the flame itself is
not known in the calculation. Even if it were, it would be difficult to model its effect on the
emission of thermal radiation when using a coarse grid. All that is usually known about a
given fuel is how much of its mass is converted into soot and transported away from the
fire. Because of its dependence on the soot volume fraction and on the temperature raised
to fourth power, the source term in the RTE must be modeled in those grid cells cut by the
flame sheet. Away from the flame, where the temperatures are lower, greater confidence in
the computed temperature and soot volume fraction permits the source term to take on its
traditional value. Thus, in FDS a decision is made when computing the source term in the
RTE based on whether combustion is occurring in a given grid cell
κ σ T 4 /π Outside flame zone
κ Ib =
(12)
χr q̇000 /4π Inside flame zone
where q̇000 is the chemical heat release rate per unit volume and χr is the local fraction of
that energy emitted as thermal radiation. Note the difference between the prescription of
a local χr and the resulting global equivalent. For small fires (D < 1 m), the local χr is
approximately equal to its global counterpart, however, as the fires increase in size, the
global value will typically decrease due to a net re-absorption of the thermal radiation by
the increasing smoke mantle [9].
LARGE SCALE FIRE SIMULATIONS
To date, the new combustion and radiation solvers have been applied to a wide variety
of fire problems to assess the cost, robustness and accuracy of the new routines. One
of the first concerns for both sub-models was their cost. The mixture fraction combustion
algorithm requires the solution of an additional transport equation for Z, adding about 20 %
to the overall CPU time. The radiation routine could cost an unacceptably high amount if
the gray gas assumption was not applied, and if the entire RTE were solved every time
step. Since radiation accounts for about 35 % of the energy transport in a typical fire
scenario, it was decided that no more than 35 % of the CPU time ought to be devoted to the
radiation transport. As a cost-saving measure, the gray RTE equation is solved gradually
over approximately 15 time steps. Every 3 time steps 1/5 of the approximately 100 solid
angle equations are updated, and the results stored as running averages. Although the code
user can control these parameters, it has been found that with the given defaults, the finite
volume solver requires 15 % to 20 % of the total CPU time of a calculation, a modest cost
given the complexity of radiation heat transfer.
A calculation is presented here that are typical of the type of fire scenario that the model has
been re-designed to address. A snapshot from the FDS visualization package Smokeview
is shown in Fig. 2. A small cushion is ignited on a couch in a room that is roughly 5 m
by 5 m by 2.5 m with a single door leading out. The fire grows to the point of flashover
in about 3 min. This simulation is based on an experiment performed by the University of
Maryland and the Bureau of Alcohol, Tobacco and Firearms.
The new combustion and radiation routines are crucial to this calculation because towards
832
FIGURE 2: Sample simulation of a room fire using the new combustion and radiation
routines. Shown is the flame sheet where the mixture fraction is at its stoichiometric
value.
flashover and beyond, the room conditions are severely underventilated and radiation is
the dominant mode of heat transfer. The gray gas assumption is made because the radiation is dominated by soot, and because the relative coarseness of the numerical grid
(10 cm) does not justify the expense of the multi-band radiation model. The fuel consists
of polyurethane, wood, and a variety of fabrics whose thermal properties are known only
in the most general sense. The soot volume fraction is based solely on estimates of the
smoke production; the actual values within the flames are unknown. In generating effective absorption coefficients with RadCal, it is assumed that the fuel is methane. Clearly
more research is needed to fill in many of the missing pieces. Refinement of the numerical
algorithm and comparison with experiment is ongoing.
GRID DEPENDENCE
During a period of about a year when the combustion and radiation models were being
implemented and tested, the effect of the numerical grid on the results was a major concern. Now that the emphasis had shifted from smoke movement away from the fire to heat
transfer in the immediate vicinity of the fire, the size of the numerical grid cells became
833
more important. In many calculations that involve either relatively large spaces or relatively small fires, the grid resolution in the vicinity of the fire will be severely limited. For
many applications, this in itself may not be a problem since the fire merely serves as a
point source of heat and smoke. However, if one is interested in flame spread, near-field
heat transfer is all-important, and the resolution of the numerical grid, especially during the
early stage of a fire, cannot be ignored.
We have found from various validation exercises involving pool fires and small compartments that good agreement with experimental data is possible when the fire is adequately
resolved. However, even in cases where the fire is not well-resolved, it is possible to get
a reasonable approximation of the total and radiative heat release rate of the fire, plus its
volumetric distribution and flame height, with numerical grids that are very coarse. To do
this, one needs to slightly modify the procedure for obtaining the local heat release rate
from the mixture fraction field. Note that this approximation is only intended for underresolved fires. The above procedure for determining the local heat release rate works well
for calculations in which the fire is adequately resolved.
What do we mean by “adequately resolved”? It depends on what the objective of the
calculation is. To a chemical kineticist, adequate resolution might involve micrometers
and microseconds; to a fire dynamicist, millimeters and milliseconds; to a fire protection
engineer, meters and seconds. A relative measure of how well a fire is resolved numerically
is given by the nondimensional expression D∗ /δx, where D∗ is a characteristic fire diameter
∗
D =
Q̇
√
ρ∞ c p T∞ g
25
(13)
and δx is the nominal size of a grid cell. Note that the characteristic fire diameter is related
to the characteristic fire size via the relation Q̇∗ = (D∗ /D)5/2 , where D is the physical
diameter of the fire. The quantity D∗ /δx can be thought of as the number of computational
cells spanning the characteristic (not necessarily the physical) diameter of the fire. The
more cells spanning the fire, the better the resolution of the calculation. For fire scenarios
where D∗ is small relative to the physical diameter of the fire, and/or the numerical grid is
relatively coarse, the stoichiometric surface Z = Z f will underestimate the observed flame
height [4]. It has been found empirically that a good estimate of flame height can be found
for crude grids if a different value of Z is used to define the combustion region
Z f ,eff
D∗
(14)
= min 1 , C
Zf
δx
Here C is an empirical constant to be used for all fire scenarios. As the resolution of
the calculation increases, the Z f ,eff approaches the ideal value, Z f . The benefit of the
expression is that it provides a quantifiable measure of the grid resolution that takes into
account not only the size of the grid cells, but also the size of the fire.
A practical consideration when implementing this idea in the numerical model is that in
most cases D∗ is not known in advance if the fire is allowed to spread throughout a space.
Somehow the quantity D∗ /δx must be approximated based only on values of the fuel mass
834
FIGURE 3: Simulations of a 0.4 m by 0.4 m sand burner with progressively coarser
resolution; from left to right, 2.5 cm, 5 cm, 10 cm and 20 cm grid cells.
flux, cell size, and mixture fraction near the burning surface. Assuming the actual fire
diameter D ∼ n δx where n is the number of cells spanning the fire, it can be shown after
substituting terms that
D∗
∗ 2/5
∼ n4/5 Q̇local
(15)
δx
where Q̇∗local is a quantity that resembles Q̇∗ , but it is defined locally
q̇00
p
Q̇∗local =
ρ∞ c p T∞ g δx
(16)
The number of cells spanning D∗ , n, is not readily obtained in a calculation, but it has been
found from numerous trials that n is proportional to the maximum value of Z in the gas
phase cells of the numerical grid. In most practical calculations, Z is far less than its ideal
value of unity in the first gas phase grid cell above the burner due to the increased numerical
diffusion necessitated by the coarse grid.
Figure 3 shows the flame sheet from four simulations of a simple 0.4 m by 0.4 m propane
sand burner set to 160 kW. The only difference between each is the grid resolution. Only
the case with a 2.5 cm grid cell was run without need of the modified flame surface value.
For the 5 cm case, the resolution factor was about 0.7, for the 10 cm case, the factor was 0.3,
and for the 20 cm case the factor was about 0.1; all approximate since the factor fluctuates
slightly during the calculation. For propane, Z f = 0.06. Since the diffusion coefficient used
in the calculation is only of order δx2 , the flame surface in the 20 cm case would just barely
appear above the burner. An other way to look at it is that we are assuming fuel and oxygen
burn instantaneously. The mixture fraction combustion model is equivalent to tracking
835
propane and oxygen through their respective transport equations and never allowing both
fuel and oxygen to exist in a single grid cell. As fuel emerges from the burner surface, it will
mix with a disproportionately large amount of oxygen in the first grid cell adjacent to the
burner. All of the heat of combustion will be liberated in that first grid cell, and the flame
height will be under-predicted. This phenomenon was noticed by Ma and Quintiere [4]
who were doing some validation exercises involving flame height.
The adjustment of the flame surface is almost always necessary when the simulation starts
with a small ignition source. It is usually impractical to provide a fine grid wherever the fire
resides, since usually the fire will grow and spread. The benefit of the technique described
here is that as the fire grows, D∗ grows, and the reliance on the adjusted flame sheet value
diminishes, and in many cases goes away entirely. For example, a flashed over room from
a numerical point of view is a fire resolved with a numerical grid spanning the length and
width of the room. However, during the initial growth stage, the fire may be supported by
just a few grid cells spanning the width of the small fire. Of course, there are techniques
used in various fields of CFD to apply fine grids to where they are needed, even adjusting
the grids during a calculation. This latter technique, adaptive grid refinement, is difficult
to apply and increases the cost of the calculation significantly. A more modest technique,
known as multi-block, allows the user to specify grids of various refinement throughout the
computational domain. Its practical application would be to finely grid a room of origin in
a building, and use a coarser grid elsewhere. Such an effort is underway at NIST, but no
matter how well it works, there will probably always be a need to model a relatively small
fire on a relatively coarse grid.
Figure 4 displays some predicted flame heights versus two experimental correlations. The
calculations were performed for a 0.3 m square gas burner with no lip and heat release rates
varying from 7 kW to 1,800 kW. For the low Q∗ fires, the Cox and Chitty data is appropriate
since they used square gas burners in their experiments [10]. The adjustment of the flame
surface was used for the cases in which Q∗ < 1. For all the rest, the flame sheet location
was chosen based solely on the ideal stoichiometry. In all cases, a uniform grid was used
to simulate the fires, with a cell size of 3.75 cm or 8 cells spanning the burner width. For
the larger Q∗ fires, the numerical resolution is adequate to capture the gross features of the
fire, in particular its flame height. For the low Q∗ fires, the flame heights are comparable to
the size of a single grid cell, in which case it is not possible to accurately predict the flame
height. Even so, the energy from the fire is deposited onto the numerical grid as well as can
be expected given the numerical resolution.
CONCLUSION
New combustion and radiation routines have been implemented in a large eddy simulation
fire model. With both, we seek to improve the physical models so that we can understand
better the dynamics of a fire, but we want the model to remain useful for practicing engineers who have limited computing resources and limited information about the materials
burning in a typical fire. Even as we develop more sophisticated numerical algorithms to
describe the growth and suppression of fires, many users will still use the model primarily
for smoke movement and detector activation in which case only the overall mass, momen836
FIGURE 4: Comparison of predicted flame heights
versus Heskestad’s correlation (solid line) [11] and
Cox and Chitty’s (dashed line) [10].
tum and energy transport from the fire is needed. The objective of the model development
is to make it as useful as possible for a wide variety of applications, both fundamental
and applied. If successful, the model will be exercised by a large variety of users which
eliminates computer bugs, introduces field modeling to a new generation of fire protection
engineers, and most importantly provides validation of the algorithms for a wide variety of
experimental data sets. This is an evolving process. Work still needs to be done in many
areas, especially soot production, radiation absorption coefficients, numerical resolution,
and material properties. Progress will be made because of better research in the future, but
also because of lessons learned from calculations being performed now. The development
of zone models over the past few decades certainly benefited from the wide use of the early
models, pointing out both the strengths and weaknesses, and ultimately guiding the development of newer algorithms. The same will be true of field models if the proper balance is
struck between research and practical application.
ACKNOWLEDGEMENTS
The authors would like to thank Drs. Paul Fuss and Anthony Hamins for their assistance
with the radiation absorption coefficients.
837
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